Moreover, from Lemma 1 of [1]}, \(D_f\)  can be expressed in terms of its lower bound via Fenchel convex duality
\(\small D_f(c_U||\pi _U) \ge \sup _{T\in \mathbb {R}} \biggl \lbrace  \mathbb {E}_{\mathbf {u} \sim c_U(\mathbf {u})} \bigl [T(\mathbf {u})\bigr ]-\mathbb {E}_{\mathbf {u}\sim \pi _U(\mathbf {u})}\bigl [f^*\bigl (T(\mathbf {u})\bigr )\bigr ]\biggr \rbrace ,\) 
